Flutter, limit cycle oscillation, bifurcation and stability regions of an airfoil with discontinuous freeplay nonlinearity

摘要

This paper is devoted to study a two-dimensional airfoil oscillating in pitch and plunge degrees of freedom. A nonlinear analysis is performed to investigate the effects of a discontinuous freeplay nonlinearity in pitch on the response of the airfoil system. In fact we show that in the presence of freeplay, the air velocity has a direct effect on the pitch vibrations of the airfoil system. Namely, it can generate the flutter leading to the limit cycle oscillation for the airfoil. With the aid of a fixed point of the Poincar map of the system and numerical findings, we determine the flutter and the limit cycle oscillation of that. The frequency, period of the limit cycle oscillation of pitch motion and the flutter speed are calculated. Tangent points are also computed, and it is shown that these points cannot be two-fold singularities for the system. Furthermore, by using the theoretical techniques of discontinuous systems, we will obtain parametric regions for the existence of grazing bifurcation (global bifurcation). The existence of grazing bifurcation helps us to display that for some values of the air velocity, different transitions or sudden jumps can occur in the system's response. Numerical results demonstrate that these transitions are accompanied by the appearance and disappearance of a tangential contact between the trajectory and the switching boundaries. Also they can cause a change in the response of the pitch motion from simply periodic to double periodic (periodic-2). Moreover, stability regions for the airfoil system with freeplay will be found. The property of these stability regions is that inside them there exist no flutter and limit cycle oscillation. Some numerical examples are given which are in good agreement with our theoretical results.